Apparatus and method for measuring decay in intensity of electromagnetic radiation in multipass spectrometry

ABSTRACT

An apparatus for measuring decay in intensity of electromagnetic radiation passing through a radiation-absorbent sample due to absorption of radiation by the sample is disclosed which includes a source of electromagnetic radiation having a wavelength within an absorption band of the sample and a plurality of partially-reflective specular surfaces which are spaced apart from each other along a predetermined path through the sample, each specular surface separating the incident radiation into a reflected part which follows the predetermined path and an unreflected path, the value of the decay being derived from intensity measurements of the unreflected parts made at different positions along the predetermined path.

FIELD OF THE INVENTION

This invention relates to an apparatus and method for measuring decay inintensity of electromagnetic radiation passing through aradiation-absorbent sample due to absorption by the sample.

The apparatus and method can be used, inter alia, to obtain a value of asample a parameter, such as the concentration of absorbent atoms ormolecules related to the decay in intensity.

BACKGROUND OF THE INVENTION

Absorption Spectroscopy has long been a tool in the analytical chemists'repertoire of analytical techniques. The fundamental theory is based onthe observation that atoms and molecules absorb energy fromelectromagnetic radiation of a particular wavelength, typically, but notexclusively in the form of photons passing through a sample. As eachatom or molecule has a distinctive pattern of absorption wavelengths itis possible to deduce the atomic or molecular species under analysis. Itis also possible to obtain a measure of the concentration of the speciesby means of reference to a calibration. The Beer-Lambert Lawquantitatively relates the reduction in intensity of electromagneticradiation of a particular wavelength to the concentration of absorbers(atoms or molecules) of that wavelength and the distance travelled bythe electromagnetic radiation through the sample, by the expression:I _(d) =I _(o) e ^(−Nβd)  Eq ^(n) 1where N=the number of absorbers per unit volume.

-   -   β=the absorption coefficient.    -   I_(o)=the initial intensity of the source.    -   I_(d)=the measured intensity after passing through the sample.    -   d=the distance travelled through the sample.

This can be rearranged as follows: $\begin{matrix}{N = {\frac{1}{\beta\quad d}{{Ln}\left( \frac{I_{o}}{I_{d}} \right)}}} & {{Eq}^{\underset{\_}{n}}2}\end{matrix}$Conventional Absorption Spectroscopy has a number of limitations.Although theoretically very sensitive and potentially quantitative,practical limitations imposed by current instrumentation and the methodof measurement seriously restrict these capabilities and hence limit theapplication of absorption spectroscopy. One of these limitations issensitivity of the measurement technique. From analysis of eqn. 1 it canbe shown that there is an optimal range of values for the exponentialterm βNd. It is then clear that for a given value of β and in order tomaintain the optimal value of the expression a reduction of the numberof absorbers N must be compensated for by a corresponding increase inthe value of d. Further analysis shows that for very small values of N,d can become impractically large. Long path lengths are achieved bymeans of folded geometries which are described in more detail later. Itis also possible to deduce an effective dynamic range for a giveninstrumental configuration from eqn 1.

Further practical limitations in electromagnetic source stability overboth short and medium term time scales and limitations in detectionsystems impose further limitations on sensitivity and accuracy ofmeasurements. From Eqn 2 it can be demonstrated that there is a need forhigh accuracy of measurement in both I₀ and I_(d). Since radiationsources have limited stability as stated previously, for the mostaccurate measurement both I₀ and I_(d) must be measured simultaneously.This introduces additional cost and complexity to instrumentation. Bothlimitations described have partial solutions that are implemented incurrent instrumentation.

Further limitations become apparent in issues of quantification andreproducibility of measurements. These limitations arise from acombination of some of the previously described practical limitationsand from the two point measurement system using I₀ at the source andI_(d) at the end of the propagation path through the sample.

Many techniques have been devised with the objective of improving theperformance of absorption spectroscopy. These include the use of foldedpath analysis cells that extend the optical path through the sample andthereby improve sensitivity, the use of detector arrays forspectroscopic analysis and the use Cavity Ringdown Laser AbsorptionSpectroscopy.

A paper by J. U. White in J. Opt. Soc Am, Vol 32, pp 285-288 describes ageometrical arrangement for extending the pathlength of a light sourcethrough an absorption cell by use of multiple reflections and traversalsthrough the sample volume, thereby achieving an extended optical path ina small volume. This type of geometry has been widely adopted inabsorption spectroscopy and is referred to as either a White cell ormore generically as a folded path geometry cell. FIG. 1 of theaccompanying drawings shows a typical folded path cell geometry. A lightsource 11 produces a light beam 12 which enters the cell 13 through awindow 14 that is optically transparent at a wavelength of interest. Thelight beam 12 within the cell 13 is allowed to traverse the cell untilit is incident upon mirror 15 where it is focused and redirected as abeam 16 towards a second optically transparent window 17 and thence tothe detector 18. The main advantage of this geometrical arrangement isto increase the pathlength through the cell thereby increasing thesensitivity of the absorption measurement, as previously discussed. Manyvariations on this type of geometry have been devised including multiplereflection types. All have at least one common feature; that is,detection is carried out at the end of the propagation path of the lightbeam through the sample.

Another example of a multiple-pass cell is described in U.S. Pat. No.5,220,402. As shown in FIG. 2 of the accompanying drawings, thisarrangement comprises a focused light source 21 that is directed into acircular cell 22 through a wedge-shaped lens 23 that also acts as awindow into the cell. The internal structure of the cell 22 is such thatit provides some focusing of the light in both axial and radial planesgiving a focal point close to the axis of the cell. By way ofillustration, FIG. 2 shows one of a plurality of paths that the lightmight follow inside the cell. Light entering the cell through the lens23 is directed to be incident on a reflecting surface 24 of the cellwhere it undergoes a reflection 25 directing it back through the axialregion of the cell to a farther reflection point on the housing wall.This process continues until the light exits the cell through a furtherlens 26 where it is directed to a means of measurement 27. The primaryaim is to maximise the optical path length within a cell of givenvolume, and both two and three dimensional geometries are described.

U.S. Pat. No. 5,485,276 describes another arrangement for increasing theoptical path length of light within a gas absorption cell using multiplereflections and a collimated beam. As shown schematically in FIG. 3,this arrangement comprises a diode laser 30 producing a collimated beam31 which enters a gas absorption cell 32 through an aperture or window33. The collimated beam 31 inside the cell 32 is so directed that it isincident upon a mirror 35 on the opposite side of the cell 32. Themirror is angled so that the reflected beam 36 is then directed towardsa further mirror 37 positioned on the same side as the entranceaperture, but displaced from it. During traversal between mirrors, thelight suffers a loss in intensity due to absorption by absorbing fluidin the cell. Further reflections and traversals continue in a likemanner until the light exits the cell at an exit aperture 38 whereuponit is detected by a detector 39. The mirrors are all contained within atwo dimensional plane.

In general, these folded path geometries have been used for the purposeof increasing the path length of the light through a sample in order toimprove measurement capability for low concentrations in absorbancespectroscopy.

It is also known to use multiple detectors in spectroscopy, primarily indetector arrays used for measuring spectra. For example, U.S. Pat. No.5,721,430 describes an apparatus that uses multiple detectors in an NDIRanalayser and a wideband light source. These multiple detectors areconfigured to operate in parallel at the end of the optical pathlength,and the detectors are provided with individual optical bandpass filtersenabling them to detect different wavelengths.

Another arrangement using multiple detectors at the end of thepropagation path through the sample is described in U.S. Pat. No.5,854,684. In general, multiple detectors have been used for the purposeof measuring different wavelengths in a spectroscopic instrument aftereither an optical filter system or the application of a wavelengthdispersive device.

A recent development in the field of high sensitivity opticalspectroscopy is Cavity Ringdown. The technique of Cavity Ringdown isderived from the high finesse optically resonant cavities used in lasertechnology and uses the principle of a highly resonant optical cavity inthe measurement of low concentrations of gases. In Cavity Ringdown thelaser light is introduced into the cavity from an external laser source.Cavity Ringdown has its roots in a paper by J. M. Herbelin et alpublished in the J. Appl. Opt. 19(1) p 144-147 in 1980 in which hedescribes the measurement of the reflectivity coefficient of highperformance mirrors using a cavity attenuated phase shift (CAPS)technique. The mirrors being measured form the ends of the opticallyresonant cavity. In the described CAPS technique the cavity decay time(deduced from the induced phase shift) is used to calculate thereflectivity coefficients of the mirrors. In the paper, concerns wereraised about the potential distortion of the calculations of thecoefficients by small quantities of absorbing contaminant gases in theoptically resonant cavity. It was further noted that the CAPS techniquemay have application in the measurement of the concentrations of smallquantities of gas deliberately introduced into the optically resonantcavity. A paper by D. Z. Anderson et al published in Appl. Opt. 23(8) p1238 in 1984 describes a further development in which the decay of lightintensity with time is directly observed.

The advent of high performance, fast pulsed lasers obviates the need forattenuated phase shift measurements, and a paper by A. O'Keefe et al inRev Sci Inst, 1988, 59(12), p2544 describes the use of pulsed lasers andthe measurement of decay in intensity with time of the pulseintracavity. All subsequent developments in cavity ringdown techniquesare based on variations of this basic approach. A review of the work ofHerbelin et al and of the subsequent developments is contained in apaper by J. J. Scherer et al in Chem Rev 1997, v 97, pp25-51.

A typical cavity ringdown apparatus, is now described by reference toFIGS. 4 a and 4 b. The apparatus consists of laser 40, a high finesseresonant optical cavity 41, a photon mulitplier 42, an amplifier 43, anoscilloscope 44 and a computer 45. The optical cavity consists of anouter housing 46, typically a cylinder, and two high reflectivityconcave mirrors 47,48 that are additionally used to seal the ends of thecavity 41. The mirror 47 nearest the laser 40 is called the entrancemirror and the mirror 48 at the other end the exit mirror. The mirrors47,48 have a coefficient of reflectivity, R, which is typically of theorder of 0.995 or higher. The cavity contains an absorbant gas speciesfor analysis by absorption of photons.

The oscilloscope 44 is used to digitise the amplified signal from thephoton multiplier 42 and the computer 45 is used for general control ofthe timing electronics and for recording the digitised output from theoscilloscope.

In a typical cavity ringdown experiment, a fast (of the order ofnanoseconds) pulse of laser energy of a known wavelength is focused anddirected into the optical cavity through the entrance mirror 47. A smallamount of the laser energy equal to (1-R), is coupled into the cavityand the rest of the laser energy is reflected back from the mirror andhas no further function in the measurement. The light in the cavity isnow trapped and reflects back and forth between the two mirrors 47,48. Asmall fraction (1-R) of the trapped laser energy passes through eachmirror at each reflection. By measuring the small component (1-R) of thelight that is transmitted through the exit mirror 48 after eachtraversal through the cavity as a function of time t, a measure of thedecay of the light pulse can be made. This decay with time I(t) is due acombination of reflection losses and absorption by the gas contained inthe cavity. This measured intensity can be shown to be proportional tothe losses in the cavity where,I(t)∝R _(tot) e ^(−σ(λ)Nt)  Eqn3Equation 3 has the form of the Beer-Lambert law which relates the lossesdue to absorption to the number of absorbers present in the cavity, butis modified to allow for the additional losses due to multiplereflections, where R_(tot) is the total loss coefficient due to thereflections. A typical decay curve is shown in FIG. 4 b. By calibratingthe apparatus with no absorbing gas present, a value for R_(tot) duesolely to reflection at the mirrors can be determined. This value canthen be taken into account in the measurement of the decay when anabsorbing gas is present, and the number of absorbers determined. Decaytimes for a typical cavity ringdown measurement are usually in theregion of the one to tens of microseconds, in part dependent upon thecavity length and also on the concentration of absorbers present.

A drawback of the cavity ringdown technique is the requirement tomeasure the decaying intensity of electromagnetic radiation as afunction of time, typically over a time interval which is only of theorder of tens of microseconds. Accordingly, implementation of cavityringdown techniques is both complex and expensive, requiring the use offast pulsed lasers, high finesse, high Q optical cavities and high speeddigital timing electronics.

The cavity ringdown technique also has the disadvantage that the pulsedelectromagnetic radiation undergoes multiples passes through the samesample region, and this may reduce the sensitivity of the measurementbeing made.

Another major problem associated with the use of cavity ringdown is thepoor efficiency of coupling of the electromagnetic radiation into theoptical cavity. U.S. Pat. No. 5,815,277 describes a method foralleviating this problem. In this method, an acoustic optical modulator(AOM) is placed inside the cavity in order to redirect the light pulseonto the optical axis of the resonant cavity. This increases thecoupling efficiency to in excess of 40%. Although this approach givessignificant improvements in efficiency, it does introduce an additionaloptical component into the cavity which may decrease the effectivenessof both the measurement and its applicability. The AOM adds to the costand to the complexity of the control system, factors which must beconsidered against a reduction in the requirements for the initialenergy of the laser source and the potential to use lower cost detectionmethods.

A further recent development in the use of cavity ringdown is atechnique known as Intracavity Laser Spectroscopy (ILS) where the lasercavity itself is used as the analysis cell. This type of device has manyadvantages in terms of size and sensitivity, but is still limited in itsapplicability. U.S. Pat. Nos. 5,841,533, 5,742,054 and 5,747,807describe three examples of the application of this technique.

It is an object of the present invention to provide an apparatus andmethod for measuring decay in intensity of electromagnetic radiationpassing through a radiation-absorbent sample due to absorption by thesample which at least alleviates the above-described short-comings ofexisting arrangements.

SUMMARY OF THE INVENTION

According to one aspect of the present invention there is provided anapparatus for measuring decay in intensity of electromagnetic radiationpassing through a radiation-absorbent sample due to absorption ofradiation by the sample, comprising a source of electromagneticradiation having a wavelength within an absorption band of the sample,partially-reflective means for partially reflecting said electromagneticradiation at successive positions which are spaced apart from each otheralong a predetermined path through the sample, said partially-reflectivemeans being effective at each said successive position to separateincident radiation into a reflected part which is caused by thepartially-reflective means to follow said predetermined path and anunreflected part, and derivation means for deriving a value of saiddecay from measurements of intensity of the unreflected parts of theelectromagnetic radiation produced at a number of different saidpositions along said predetermined path.

According to another aspect of the invention there is provided a methodfor measuring decay in intensity of electromagnetic radiation passingthrough a radiation-absorbent sample due to absorption of radiation bythe sample, comprising, generating electromagnetic radiation having awavelength within an absorption band of the sample, partially-reflectingsaid electromagnetic radiation at successive positions which are spacedapart from each other along a predetermined path through the sample,whereby to separate the radiation into a reflected part which is causedto follow said predetermined path and an unreflected part, and derivinga value of said decay from measurements of intensity of the unreflectedparts of the electromagnetic radiation produced at a number of differentsaid positions along said predetermined path.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention are now described, by way of example only,with reference to the accompanying drawings of which:

FIG. 1 is a schematic representation of a known, folded-path geometryabsorption cell,

FIG. 2 is a schematic representation of a known multiple-pass absorptioncell,

FIG. 3 is a schematic representation of another known gas absorptioncell,

FIG. 4 a is a schematic representation of a known cavity ringdownapparatus,

FIG. 4 b shows a typical decay curve obtained using the apparatus ofFIG. 4 a,

FIG. 5 is a schematic representation of a measuring apparatus accordingto the invention,

FIG. 6 shows another embodiment of a measuring apparatus according tothe invention,

FIG. 7 is a schematic representation of a specular surface in themeasuring apparatus of FIG. 6,

FIG. 8 a shows a decay curve obtained using the apparatus of FIG. 6,

FIG. 8 b shows a further curve obtained from the decay curve of FIG. 8a, and

FIGS. 9, 10 and 11 are diagrams useful in understanding the derivationof certain mathematical expressions referred to in the description.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 5 shows a schematic representation of an embodiment of theinvention.

The apparatus shown in FIG. 5 comprises two parallel, flat,partially-reflective mirrors 51,52 having a configuration akin to aFabry Perot resonator. A source 53 directs a beam 54 of electromagneticradiation onto the facing surface of mirror 52. The beam is incident onthis surface at a small angle (typically less than 10°) with respect tothe normal and so undergoes multiple reflections between the facingsurfaces of the mirrors, as shown in the drawing.

With this arrangement, the beam 54 is constrained to follow apredetermined, extended path between the mirrors, being alternatelyreflected at each mirror at successive positions spaced apartequidistantly from each other along the path.

A number of detectors is provided to measure the unreflected parts ofthe electromagnetic radiation at different positions D along theextended path followed by the beam. As will be described in greaterdetail hereinafter with reference to FIG. 6, these measurements can beused to derive a value of decay of intensity of electromagneticradiation due to absorption by a radiation-absorbent sample containedbetween the mirrors 51,52. As will be explained, because the path lengthd between successive positions D can be accurately measured, a value ofdecay due to absorption can be derived without any need to measureintensity as a function of time t, although the timing of eachmeasurement can still be calculated by application of the simplerelationship t=d/nc, where n is the refractive index of the sample and cis the speed of light in vacuum.

Therefore, in comparison with known cavity ringdown arrangementsdescribed hereinbefore, which do rely upon time-based measurements, theapparatus of FIG. 5 offers reduced complexity and cost, and does notrequire fast timing electronics which may be susceptible to timingjitter. In contrast, the apparatus of FIG. 5 relies on the accuratemeasurement of path length of between successive detection positions D.

FIG. 6 shows another embodiment of the invention. Referring to thisFigure, the measuring apparatus includes a chamber 61 enclosing a sampleregion 62, and a plurality of inwardly-facing, partially-reflectivespecular surfaces S₀, S₁ . . . S₉ supported by, or forming part of, thechamber wall. For clarity of illustration, only ten such surfaces areshown. The chamber 61 may include an inlet unit (not shown) forcontrollably admitting sample to the sample region 62 and an outlet unit(not shown) for controllably discharging sample from the sample region.

The apparatus also includes a source 64 of pulsed electromagneticradiation and an optical element 65 for focusing and calibrating thepulses causing them to enter the chamber 61 at a desired angle ofincidence α via a radiation transparent window 66.

In this particular embodiment, the source 64 is a laser producing amonochromatic beam of light. However, it will be appreciated that anyother suitable source of electromagnetic radiation may be used, and suchsources may include single monochromatic, multiple monochromatic or wideband sources that are either static or wavelength scanned. It is alsoenvisaged that the source may be able to produce. wavelengths in therange of 2 nm to 10 mm, dependent on the absorption bands of the sample.

Furthermore, in this particular embodiment the means of measurement ofthe decay in intensity of the electromagnetic radiation is by photondetectors used in a single wavelength detection mode. However it willagain be appreciated that any suitable detectors for the wavelength(s)used may be usefully employed and additionally said detectors may becombined with some wavelength dispersive or discrimination device toprovide a spectroscopic output.

Additionally, this method of measurement may also be applied to otherspectroscopic techniques which require the manipulation of the inputelectromagnetic radiation such as Fourier transform or other liketechniques.

The specular surfaces S₀, S₁ . . . S₉ are divided into two groups S₀, S₂. . . S₈; S₁, S₃ . . . S₉ arranged along two parallel lines L₁, L₂.

The relative spacings and orientations of the specular surfaces and theangle of incidence α of the pulses entering the chamber 61 via window 66are so chosen that the pulses undergo multiple reflections, causing themto follow a predetermined path through the sample region 62. Thus, asshown in FIG. 6, a pulse incident at specular surface S₀ is reflectedback through the sample region 62 to impinge on specular surface S₁which, in turn, reflects the incident pulse back through the sampleregion to impinge in specular surface S₂, and so on, until, eventually,the pulse exits the chamber 61 via an exit window 67 or its intensityfalls to zero.

In this manner, an incident pulse of electromagnetic radiation follows apredetermined path having an extended, folded configuration, passingback and forth through different parts of the sample region, givingimproved sensitivity.

Referring now to FIG. 7, each specular surface S_(r) is effective toseparate the incident photon flux J_(r) ⁻(λ) into a reflected part J_(r)⁺(λ) and a non-reflected part T_(r)(λ), where λ is the wavelength of theincident photons.

As will now be explained, the intensity of an incoming pulse (i.e. thephoton flux) progressively decays as it passes through the sample region62. This decay is attributable to two different effects; firstly, asalready described, each specular surface separates an incident pulseinto a reflected part and a non-reflected part and so the intensity ofradiation in the pulse will be reduced at each reflection; and secondly,radiation in the reflected part of the pulse will be progressivelyabsorbed by the radiation-absorbent sample through which it passes.

The relative spacings d of the specular surfaces along the predeterminedpath are precisely known and this can be exploited to derive a value ofdecay of intensity attributable to the second, sample-absorbent effect.This is accomplished by measuring the non-reflected parts of theradiation produced at some or all of the specular surfaces. To this end,a suitable detector is provided behind each specular surface where ameasurement is to be made. For clarity of illustration, only two suchdetectors 68,69 are shown in FIG. 6, located behind specular surfacesS₀, S₁ respectively.

It will be clear from FIG. 7, that the non-reflected part of theincident photon flux T_(r)(λ) (at the r^(th) specular reflector) isgiven by the expression:T _(r)(λ)=J _(r) ⁻(λ)(1−R(λ))  Eq4where R(λ) is the wavelength-dependent reflection coefficient of thereflective surface;and that the reflected part J_(r) ⁺(λ) is given by the expression:J _(r) ⁻(λ)=J _(r) ⁻(λ)R(λ)  Eq5Between the r^(th) and the (r+1)^(th) specular surfaces the reflectedpart of the photon flux will undergo a reduction due to absorption bythe sample, and it can be shown that this reduction is governed by theBeer-Lambert relationship, given by the general expression:J _(r+1)(λ)=J _(r)(λ)e ^(−σ(λ)Nd)  Eq6where N is the total number of absorbers per unit volume at a givenwavelength λ, d is the path length between specular surfaces and σ (λ)is the wavelength dependent absorption cross-section.

In the following analysis, an index k (=0, 1, 2 . . . ) is used todenote each successive detector which is provided behind a specularsurface to measure the photon flux of the unreflected part of theelectromagnetic radiation. Detectors may be provided behind all thespecular surfaces, and the index k of each detector would then be thesame as the index r of the associated surface. However, detectors may beprovided behind some, but not all of the reflectors and, in this case,the index k of a detector may not be the same as the index r of theassociated surface, subject to the initial condition that r=k=o.

It is possible to show that the reflected photon flux J_(r) ⁺(λ) at ther^(th) specular surface is given by the expression:J _(r) ⁺(λ)=J ₀(λ)R(λ)^(r) e ^(−σ(λ)rNd),  Eq7where J₀ (λ) is the photon flux incident at the first specular surface(r=0).

It can also be shown that the unreflected part of the photon fluxJ_(k)′(λ) measured by the k^(th) detector located behind the r^(th)specular surface is also governed by the Beer-Lambert relationship,given by the general expression:J _(k)′(λ)=J ₀′(λ)R(λ)^(r) e ^(−σ(λ)rNd),  Eq8where J₀′(λ) is related to J₀(λ) by the expression: $\begin{matrix}{{J_{0}(\lambda)} = \frac{{J_{0}^{\prime}(\lambda)}{R(\lambda)}}{{L_{0}(\lambda)}\left( {1 - {R(\lambda)}} \right)}} & {Eq9}\end{matrix}$where L₀(λ) is the loss of photon flux due to transmission at the firstspecular surface (r=0). Formal derivations of the expressions given byEqus 7, 8 and 9 are given hereinafter.

It will be apparent from equ 7 above that tho function Ln (J_(k)′(λ)) islinearly related to rd and because, in this embodiment, a detector isprovided at each specular surface (i.e. k=r) the function Ln (J_(k)′(λ))is linearly related to k also.

FIG. 8 a shows the decay curve formed by a plot of J_(k)′(λ) as afunction of detector number k, and FIG. 8 b shows the corresponding plotof Ln (J_(k)′(λ)) as a function of k. The slope m of the plot shown inFIG. 8 b is given by the expression:m=Ln(R(λ))−σ(λ)Nd,  Eq 10from which the decay value σ(λ)Nd due to absorption can be readilydetermined.

Furthermore, provided the value of σ(λ) is known, the concentration ofabsorbers N can also be determined.

It will also be noted that the value of m is independent of the initialphoton flux J₀(λ). Therefore, a decay value can be derived withoutreference to the initial intensity of the electromagnetic radiation.Accordingly, the described apparatus does not suffer from the problemscaused by fluctuating source sensitivity encountered in some of theearlier systems described hereinbefore. Some of the limitationsregarding quantification and comparability are also alleviated.

As already explained the described embodiments of this invention do notrely on the relative timings of the measurements. Therefore, incomparison with the known cavity ringdown techniques which do rely ontime-based measurements, the described embodiments offer reducedcomplexity and cost, and do not need fast timing electronics which maybe susceptible to timing jitter. The described embodiments also have thefurther advantage that they do not require a high finesse resonantcavity with its attendant coupling problems; in contrast,electromagnetic radiation can be introduced into the sample region withrelative ease and high efficiencies approaching 100% using anappropriately angled beam. Furthermore, the electromagnetic radiationdoes not pass repeatedly through the same sample region; in contrast,the electromagnetic radiation follows a path extending through differentparts of the sample region, giving improved sensitivity.

The described embodiments have a folded-path geometry thereby increasingthe path length within a relatively compact sample region. An increasedpath length also gives improved sensitivity and this is particularlyadvantageous in the case of a sample having a low concentration ofabsorbers. However, the described embodiments do not rely upon a singledetector located at the end of the optical path which is used in knownfolded-path arrangements and which tends to limit the accuracy andprecision of the measurements. In contrast, the decay value is derivedfrom intensity measurements made at a number of different positionsalong the optical path, giving improved precision and an extendeddynamic range.

Although the described example demonstrates how the invention may beused to determine the concentration N of absorbers in the sample, theinvention may also be used to determine other sample parameters; forexample, sample parameters associated with variations of bonding,excitation or rotational state giving rise to an identifiable absorptionband.

Furthermore, although the described embodiments have a two-dimensional,folded path geometry, three-dimensional, folded path geometries are alsoenvisaged.

In the described embodiments, the specular surfaces are spaced apartfrom each other equidistantly, with a separation d. However, thereflectors (and their associated detectors) need not necessarily bespaced apart equidistantly. In the case of non-equidistantly-spacedreflectors the term rd in Equ 8 would be replaced by the term d_(r)representing the actual distance of r^(th) reflector along the opticalpath, and the function Ln (J_(k)′(λ)) would then be linearly related tod_(r).

There now follows derivations of Equations 7, 8, 9 and 10 above.

Derivation of Equation 7

Referring to FIG. 9, the photon flux is incident upon the area B andpasses through the volume which contains the absorbing gas species.

We define the following parameters.

-   -   J_((r))(λ) is the photon flux entering the volume. (r=0, 1, 2 .        . . )    -   J_((r+1))(λ)is the photon flux leaving the volume at z=d    -   J(z,λ)is the photon flux at any point z along the z axis between        0 and d    -   N is the number of absorbers per unit volume.    -   B is the cross sectional area of the volume.    -   σ(λ) is the effective, wavelength dependent, absorption cross        sectional area.    -   d is the path length between reflections

The number of absorbers per unit length is given by NB

We can derive the value for ΔJ as:${- {{dJ}(\lambda)}} = \frac{{J\left( {z,\lambda} \right)}\left\lbrack {{\sigma(\lambda)}{NBdz}} \right\rbrack}{B}$Therefore the ratio of absorption is given by:$\frac{{dJ}(\lambda)}{J\left( {z,\lambda} \right)} = {{- {\sigma(\lambda)}}{Ndz}}$Integrating this over the range of z for the volume gives${\int_{J_{(t)}{(\lambda)}}^{J_{({r + 1})}{(\lambda)}}{\frac{1}{J(z)}\quad{\mathbb{d}{J(\lambda)}}}} = {{- {\sigma(\lambda)}}N{\int_{0}^{d}\quad{\mathbb{d}z}}}$ Ln(J _((r+1))(λ))−Ln(J _((r))(λ))=−σ(λ)NdTaking the exponential of both sides gives: $\begin{matrix}{\frac{J_{({r + 1})}(\lambda)}{J_{(r)}(\lambda)} = {\mathbb{e}}^{{- {\sigma{(\lambda)}}}{Nd}}} & {{Eq}^{\underset{\_}{n}}11}\end{matrix}$So the photon flux at z=d is given byJ _((r+1))(λ)=J _((r))(λ)e ^(−σ(λ)Nd)  Eq ^(n) 12

Referring to FIG. 10. we can now consider the case of a reflection atz=d where

-   J_((r+1)) ⁻(λ)=The photon intensity at reflection r+1 just before    the reflection occurs.-   J_((r+1)) ⁺(λ)=The photon intensity at reflection r+1 just after the    reflection occurs.-   (in between J_((r+1)) ⁻(λ) & J_((r+1)) ⁺(λ) the losses in photon    flux due to absorption by gas is assumed to be zero).-   R(λ)=the wavelength dependent reflectivity coefficient.-   A(λ)=is the wave length dependent absorption of the sample gas-   r is the number of reflection events.

From FIG. 10 we see that $\begin{matrix}{{J_{({r + 1})}^{+}(\lambda)} = {{R(\lambda)}J_{({r + 1})}^{-}}} & \\{= {{R(\lambda)}{J_{(r)}^{+}(\lambda)}{A(\lambda)}}} & {{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}{{Eq}^{\underset{\_}{n}}13}} \\{= {{R(\lambda)}{J_{(r)}^{+}(\lambda)}{\mathbb{e}}^{{- {\sigma{(\lambda)}}}{Nd}}}} & {{Eq}^{\underset{\_}{n}}14}\end{matrix}$Equation 14 is the general expression for the decay in photon fluxcaused by a single absorption followed by a single reflection at amirror surface.

W can now expand this for multiple reflections and we see that, for rsuccessive absorptions and reflections from J₍₀₎ ⁺(λ) to J_(r+1) ⁺(λ) wehave:

 J _(r) ⁺(λ)=J ₍₀₎ ⁺(λ)[R(λ)Λ(λ)]′

We can now define J₀(λ) as being the photon flux immediately after thezeroth reflection where r=0 such thatJ ₀(λ)=J ₀ ⁺(λ)

The general expression for r absorptions and reflections can be writtenas $\begin{matrix}\begin{matrix}{{J_{r}^{+}(\lambda)} = {{J_{0}(\lambda)}\left\lbrack {{R(\lambda)}{A(\lambda)}} \right\rbrack}^{\prime}} \\{= {{J_{0}(\lambda)}{R(\lambda)}^{r}{\mathbb{e}}^{{- {\sigma{(\lambda)}}}{rNd}}}}\end{matrix} & {{Eq}^{\underset{\_}{n}}15} \\\begin{matrix}{{J_{r}^{-}(\lambda)} = {{J_{0}(\lambda)}{R(\lambda)}^{r - 1}{A(\lambda)}^{\prime}}} \\{= {{J_{0}(\lambda)}{R(\lambda)}^{({r - 1})}{\mathbb{e}}^{{- {\sigma{(\lambda)}}}{rNd}}}}\end{matrix} & {{Eq}^{\underset{\_}{n}}16}\end{matrix}$Derivation of equations 8, 9, & 10

In order to demonstrate the decay function, we need to define therelationship between the measured photon flux at detector k related tothe initial photon flux J₀(λ). Since each detector is uniquelyassociated with a reflection r (r not necessarily being equal to k) thenwe can define the following parameters and initial conditions.

-   -   Let each detected photon flux be J_(k) ⁺(λ)    -   Let each detector be k where k=0, 1, 2 . . . K        and the initial condition of r=k=0        So        J _(k) ⁺(λ)=T _(r)(λ)L _(k)J_(r) ⁻(λ)  Eq ^(n) 17        where:

L_(k)(λ)=the total transmission loss between the unreflected componentand the detector k.

And T_(r)(λ)=the total of the unreflected component of the incidentphoton flux at reflection r

Now from FIG. 11 $\begin{matrix}{{T_{r}(\lambda)} = {{J_{r}^{-}(\lambda)} - {J_{r}^{+}(\lambda)}}} \\{= {{J_{r}^{-}(\lambda)} - {{J_{r}^{-}(\lambda)}{R(\lambda)}}}} \\{= {J_{r}^{-}\left( {1 - {R(\lambda)}} \right)}}\end{matrix}$substituting for=J_(r) ⁻(λ) from Eq ^(n) 16 gives:T _(r)(λ)=J ₀(λ)R(λ)^((r−1)) e ^(−σ(λ)rNd)(1−R(λ))  Eq ^(n) 18We can now include transmission coefficients related to the k^(th)detector position.

From FIG. 11L _(k)(λ)=LO _(k)(λ)LM _(k)(λ)LD _(k)(λ)QE _(k)(λ)G _(k)(λ)  Eq ^(n) 19where:

-   LM_(k)(λ)=is the wavelength dependent transmission coefficient    through the mirror.-   LO_(k)(λ)=is the collective wavelength dependent optical    transmission coefficients related to any optical elements between    the mirror and any dispersive device or detector-   G_(k)(λ)=is the wavelength dependent dispersion characteristics of    any dispersive device. In the absence of a dispersive device,    G_(k)(λ)=1-   LD_(k)(λ)=is the wavelength dependent detector transmission    coefficients-   QE_(k)(λ)=is the wavelength dependent quantum efficiency coefficient    of the detector Substituting into Eq ^(n) 17 from 18 & 19 we obtain    the expression.     J _(k) ⁺(λ)=LM _(k)(λ)LO _(k)(λ)LD _(k)(λ)QE _(k)(λ)G    _(k)(λ)(1−R(λ)J ₀(λ)R(λ)^(r−1) e ^(−σ(λ)rNd))  Eq ^(n) 20    which can be simplified to the form:    J _(k) ⁺(λ)=J ₀(λ)L _(k)(λ)(1−R)(λ)R(λ)^(r−1) e ^(−σ(λ) rNd)  Eq    ^(n) 21

It is useful for to rearrange this to obtain the value for the parameterN. The most convenient form is to plot a graph of Ln(J_(k) ⁺(λ)) againstr.

Rearranging and taking logs of both sides we obtain the expression:$\begin{matrix}{{Ln}\left( {{J_{k}^{~\prime}(\lambda)} = {{\left( {{{Ln}\left( {R(\lambda)} \right)} - {{\sigma(\lambda)}{Nd}}} \right)r} + {{Ln}\left( \frac{\left. {\left( {1 - R} \right)(\lambda)} \right){L_{k}(\lambda)}{J_{0}(\lambda)}}{\left( {R(\lambda)} \right)} \right)}}} \right.} & {{Eq}^{\underset{\_}{n}}22}\end{matrix}$Which is an equation of the form y=mx+cWherey=Ln(J _(k) ⁺(λ)),

x=r,$c = {{{Ln}\left( \frac{{J_{0}(\lambda)}{L_{k}(\lambda)}\left( {1 - {R(\lambda)}} \right)}{R(\lambda)} \right)}\quad{and}}$ m=Ln(R(λ)−σ(λ)Nd)

So we can derive the value of N, the absorption parameter from$\begin{matrix}{N = \frac{m - {{Ln}(R)}}{{- {\sigma(\lambda)}}d}} & {{Eq}^{\underset{\_}{n}}23}\end{matrix}$We know the values R and d for a particular physical configuration andwe can calculate (or have previously experimentally determined) thevalue of σ(λ) m is found from the slope of the plot of Ln(J_(k) ⁺(λ))with r.

For the case where there is a linear relationship between r and k thenwe can define the following instrument parameter C${{Let}\quad C} = \frac{M}{K}$

-   -   where M is the total number of reflections r    -   and K is the total number of detectors.        So r can be expressed in terms of C and k        r=Ck

Substituting this into equation 22 changes the variable from r to k sothat equation 23 is modified to be $\begin{matrix}{N = \frac{\frac{m}{C} - {{Ln}\left( {R(\lambda)} \right)}}{{- {\sigma(\lambda)}}d}} & {{Eq}^{\underset{\_}{n}}24}\end{matrix}$

In a simple practical instrument this is the most convenient form to usewhere Ln(J_(k) ⁺(λ)) is plotted against the detector number k. We alsosee from the equation 24 that the measurement of the number of absorbersN is now shown to be independent of the initial light source intensity.

Referring back to FIG. 11 and equation 21 and setting r=k=0 we obtain anexpression for J₀ ⁺(λ) where $\begin{matrix}{{J_{0}(\lambda)} = {\frac{{J_{0}(\lambda)}{R(\lambda)}}{{L_{0}(\lambda)}\left( {1 - {R(\lambda)}} \right)} =}} & {{Eq}^{\underset{\_}{n}}25}\end{matrix}$

We can now normalise to the photon flux J₀ ⁺(λ) measured at the firstdetector with that measured at the k^(th) detector by dividing equation11 by equation 19 as follows.$\frac{J_{k}^{\prime}(\lambda)}{J_{0}^{\prime}(\lambda)} = \frac{{L_{k}(\lambda)}\left( {1 - {R(\lambda)}} \right){J_{0}(\lambda)}{R(\lambda)}^{r - 1}{\mathbb{e}}^{{- {\sigma{(\lambda)}}}{rNd}}{R(\lambda)}}{{L_{0}(\lambda)}\left( {1 - {R(\lambda)}} \right){J_{0}(\lambda)}}$

If the value of L_(k)=L₀ then the expression simplifies to:J _(k) ⁺(λ)=J ₀ ⁺(λ)R(λ)^(r) e ^(−σ(λ)rNd)  Eq ^(n) 26

For other applications, the photon flux can be replaced with an energyflux. This is more appropriate for the case of long wavelengthradiation.

It is also noted that this proof is generally applicable to any fluidthat substantially obeys the Beer-Lambert law.

This proof also can be extended to multiple absorptions where the termA(λ) can be expanded to include contributions from more than oneabsorber at that wavelength. e.g. for a given wavelength λ₁$\begin{matrix}{{A\left( \lambda_{l} \right)} = {\sum\limits_{1}^{n}\quad{a_{n}\left( \lambda_{l} \right)}}} & {{Eq}^{\underset{\_}{n}}27}\end{matrix}$

1. An apparatus for measuring decay in intensity of electromagneticradiation passing through a radiation-absorbent sample due to absorptionof radiation by the sample, comprising a source of electromagneticradiation having a wavelength within an absorption band of the sample,partially-reflective means for partially reflecting said electromagneticradiation at successive positions which are spaced apart from each otheralong a predetermined path along a single geometrical ray through thesample, said partially-reflective means being effective at each saidsuccessive position to separate incident radiation into a reflected partwhich is caused by the partially-reflective means to follow saidpredetermined path and an unreflected part, and derivation means forderiving a value of said decay from measurements of intensity of theunreflected parts of the electromagnetic radiation produced at a numberof different said positions along said predetermined path.
 2. Anapparatus as claimed in claim 1 wherein said derivation means derivessaid value of decay from measurements of intensity of the unreflectedparts of the electromagnetic radiation produced at all said positionsalong said predetermined path.
 3. An apparatus as claimed in claim 1wherein said partially-reflective means comprises a plurality ofdiscrete partially-reflective elements.
 4. An apparatus as claimed inclaim 1 wherein said partially-reflective means comprises at least onepartially-reflective element, the or each said partially-reflectiveelement being arranged to partially reflect said electromagneticradiation incident at a plurality of said positions.
 5. An apparatus asclaimed in claim 4 wherein said partially-reflective means comprises apair of parallel, partially-reflective plates arranged so that saidpredetermined path extends alternately between the plates.
 6. Anapparatus as claimed in claim 5 wherein said source is arranged todirect a beam of electromagnetic radiation onto a surface of one of saidplates at an angle to said surface no greater than 10°.
 7. An apparatusas claimed in claim 1 wherein said partially reflective means is soarranged that said predetermined path occupies a substantiallytwo-dimensional plane.
 8. An apparatus as claimed in claim 1 whereinsaid partially-reflective means is so arranged that said predeterminedpath occupies a three-dimensional space.
 9. An apparatus as claimed inclaim 1 including a chamber for containing said sample.
 10. An apparatusas claimed in claim 9 including means for admitting sample to anddischarging sample from, the chamber.
 11. An apparatus as claimed inclaim 9 wherein said partially-reflective means is supported by orformed in a wall of the chamber.
 12. An apparatus as claimed in claim 9wherein said source is external to said chamber.
 13. An apparatus asclaimed in claim 9 wherein said source is internal to said chamber. 14.An apparatus as claimed in claim 9 wherein said source forms part of thechamber wall.
 15. An apparatus as claimed in claim 1 wherein saidpartially-reflective means has substantially the same reflectioncoefficient at each said successive position.
 16. An apparatus asclaimed in claim 1 wherein said source of electromagnetic radiation is apulsed source.
 17. An apparatus as claimed in claim 1 wherein saidsource of electromagnetic radiation is a monochromatic source.
 18. Anapparatus as claimed in claim 1 wherein said source of electromagneticradiation is a wideband source.
 19. An apparatus as claimed in claim 1wherein said source simultaneously produces electromagnetic radiation ata number of discrete wavelengths.
 20. An apparatus as claimed in claim 1wherein said source of electromagnetic radiation produceselectromagnetic radiation in the wavelength range from 2 nm to 10 mm.21. An apparatus as claimed in claim 1 wherein said different positionsare spaced apart from each other equidistantly.
 22. A method formeasuring decay in intensity of electromagnetic radiation passingthrough a radiation-absorbent sample due to absorption of radiation bythe sample, comprising, generating electromagnetic radiation having awavelength within an absorption band of the sample, partially-reflectingsaid electromagnetic radiation at successive positions which is spacedapart from each other along a predetermined path along a singlegeometrical ray through the sample, whereby to separate radiation into areflected part which is caused to follow said predetermined path and anunreflected part, and deriving a value of said decay from measurementsof intensity of the unreflected parts of the electromagnetic radiationproduced at a number of different said positions along saidpredetermined path.